Recovering U(n)-invariant toric K\"ahler metrics on mathbb{C}mathbb{P}^n from the torus equivariant spectrum

In this note we prove that toric K\"ahler metrics on complex projective space which are also $U(n)$-invariant are determined by their equivariant spectrum i.e. the list of eigenvalues of the Laplacian together with weights of the torus representation on the eigenspaces.